144 XIII INTEGRATION
(b) Suppose that, for each s > 0, there exists a measurable set A, an integrable func-
tion g ^ 0, and an integer n0 such that, for all n^>n0, we have I \fn\ dp ^ e and
!/«(*)! ^ \d(*)\ for all xeA. Show that /is integrable and that lim f I/-/J dp,= 0.
n-*oo J
Consider the converse.
(c) Show by examples that the conditions of (a) are not sufficient and that the con-
ditions of (b) are not necessary for /to be integrable and \fdp. = lim \fn dfi.
J n-*oo J
5. If A is measurable and B is any subset of X, show that
^*(B) = ^*(B n A) + /z*(B n (X -A)).
(If /u*(B) < + oo, consider an integrable set Bx => B such that /x*(B) = ft(Bi).) Con-
versely, show that if A satisfies this condition (for all B <= X), then A is measurable
(cf. Section 13.8, Problem 3).
6. Suppose that X is compact. A bounded real-valued function / on X is said to be
continuous almost everywhere (with respect to /x) on X if the set of points of discon-
tinuity of /is negligible.
(a) Give an example of a function /which is continuous almost everywhere and such
that there exists no continuous function g which is equal to /almost everywhere.
(b) Suppose that the support of ft is equal to X. Show that a bounded real-valued
function /defined on X is equal almost everywhere to an almost everywhere contin-
uous function on X if and only if there exists a subset A of X such that X — A is
negligible and f\ A is continuous. (To show that the condition is sufficient, observe
that A is dense in X and that the lower semicontinuous extension of/| A to X is
continuous at every point of A.) Deduce that /is measurable. Show that there exists
a sequence (/,) of continuous functions on X which converges at every point of X and
whose limit is almost everywhere equal to/(cf. Section 13.11 , Problem 3).
(c) Show that a real-valued function / on R which is continuous on the right (i.e.,
such that/fc-f ) —f(x) for all x e R) is continuous except at the points of an at most
denumerable. set, and is therefore continuous almost everywhere with respect to
Lebesgue measure. (Apply Section 3.9, Problem 3 to the set An of points x e R at which
the oscillation of /is >!/«.)
7. A partition w = (A«) of a set E is said to be finer than a partition w' = (A'ff) of E if,
for each index a, there exists ft such that A« <= A^ . The partitions of E form an ordered
set with respect to this relation.
(a) Let X be a compact metric space. For each finite partition w = (Afc) of X con-
sisting of integrable sets and each bounded real-valued function /on X, put
Sw(/) - sup
k xeAk
O'Riemann sums" relative to /and the partition TZJ). Show that
and that if w is finer than w'9 then sw>(f) <; sw(f) and Sro(/) g Sw>(/).]£ hm <*g